| L(s) = 1 | + 1.73i·5-s + 4.73·7-s − 4.73i·11-s + 3i·13-s − 7.73·17-s − 6.19i·19-s + 4.73·23-s + 2.00·25-s + 7.73i·29-s + 3.46·31-s + 8.19i·35-s − 6.46i·37-s + 3.46·41-s − 4.19i·43-s + 0.928·47-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s + 1.78·7-s − 1.42i·11-s + 0.832i·13-s − 1.87·17-s − 1.42i·19-s + 0.986·23-s + 0.400·25-s + 1.43i·29-s + 0.622·31-s + 1.38i·35-s − 1.06i·37-s + 0.541·41-s − 0.639i·43-s + 0.135·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.420213011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.420213011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 4.73iT - 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 7.73T + 17T^{2} \) |
| 19 | \( 1 + 6.19iT - 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 7.73iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.46iT - 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 4.19iT - 43T^{2} \) |
| 47 | \( 1 - 0.928T + 47T^{2} \) |
| 53 | \( 1 + 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 2.53iT - 59T^{2} \) |
| 61 | \( 1 + 0.464iT - 61T^{2} \) |
| 67 | \( 1 - 6.19iT - 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 3.80T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.463686952148151182548840690120, −7.20935925038292287169033629565, −6.96114029717886922739699777491, −6.10981704686912414788076904824, −5.02120092724679078633568081370, −4.70864902531554780635776175231, −3.68287300978780384499883467874, −2.67316722674796989028401179626, −1.97411812066002956016493837606, −0.75338981724109321410667154149,
1.03501879130440422883097462298, 1.83441676756556250183372732591, 2.62332648909728417249106604780, 4.19520886639505228898143201949, 4.58313693463091628521382518240, 5.06995154128883121947548966783, 5.98140702453124586932015429941, 6.91273286806609712508488668960, 7.81616192841849591000089270957, 8.082980312765633545978078144769
